Consistent Maximum Likelihood Estimation Using Subsets with Applications to Multivariate Mixed Models

TL;DR

This paper develops a new theoretical framework for establishing the consistency of maximum likelihood estimators in multivariate mixed models using data subsets, relaxing traditional assumptions and demonstrating applicability to complex models.

Contribution

It introduces a novel subset-based approach for proving estimator consistency that requires fewer assumptions than existing methods, applicable to non-stochastic predictors and mixed-type responses.

Findings

Proves consistency of MLE in multivariate mixed models with non-stochastic predictors.

Extends subset-based consistency theory to models with mixed discrete and continuous responses.

Demonstrates the approach on models where previous consistency results were unknown.

Abstract

We present new results for consistency of maximum likelihood estimators with a focus on multivariate mixed models. Our theory builds on the idea of using subsets of the full data to establish consistency of estimators based on the full data. It requires neither that the data consist of independent observations, nor that the observations can be modeled as a stationary stochastic process. Compared to existing asymptotic theory using the idea of subsets we substantially weaken the assumptions, bringing them closer to what suffices in classical settings. We apply our theory in two multivariate mixed models for which it was unknown whether maximum likelihood estimators are consistent. The models we consider have non-stochastic predictors and multivariate responses which are possibly mixed-type (some discrete and some continuous).